Converting from Base [latex]10[/latex] to Other Bases
Converting from an unfamiliar base to the familiar decimal system is not that difficult once you get the hang of it. It’s only a matter of identifying each place and then multiplying each digit by the appropriate power. However, going the other direction can be a little trickier. Suppose you have a base-ten number and you want to convert to base-five.
How to: Convert from Base-[latex]10[/latex] to Base-[latex]b[/latex]
- Find the highest power of the base-[latex]b[/latex] that will divide into the given number at least once and then divide.
- Write down the whole number part, then use the remainder from the division in the next step.
- Repeat step two, dividing by the next highest power of the base-[latex]b[/latex], writing down the whole number part (including [latex]0[/latex]), and using the remainder in the next step.
- Continue until the remainder is smaller than the base. This last remainder will be in the “ones” place.
- Collect all your whole number parts to get your number in base-[latex]b[/latex] notation.
Remember, the remainder is not the decimal place when dividing a number. A remainder is the amount left over after dividing a number by another number. In division, the remainder is the part that’s left after dividing as evenly as possible.
For example, when you divide [latex]7[/latex] by [latex]3[/latex], you get the number [latex]2.33[/latex]. The remainder is not [latex]33[/latex]; the quotient is [latex]2[/latex], and the remainder is [latex]1[/latex], because [latex]
7=3×2+1[/latex]. Think of the remainder as what’s “remaining” after you’ve done the division!
Let’s start with a worked example before you try it on your own.
The powers of five are:
[latex]5^{0} = 1[/latex]
[latex]5^{1} = 5[/latex]
[latex]5^{2} = 25[/latex]
[latex]5^{3} = 125[/latex]
[latex]5^{4} = 625[/latex]
Etc…
Since [latex]348[/latex] is smaller than [latex]625[/latex], but bigger than [latex]125[/latex], we see that [latex]5^{3} = 125[/latex] is the highest power of five present in [latex]348[/latex]. So we divide [latex]125[/latex] into [latex]348[/latex] to see how many of them there are:
[latex]348 \div 125 = \textcolor{red}{2}[/latex] with remainder [latex]98[/latex]
We write down the whole part, [latex]2[/latex], and continue with the remainder. There are [latex]98[/latex] left over, so we see how many [latex]25[/latex]s (the next smallest power of five) there are in the remainder:
[latex]98 ÷ 25 = \textcolor{red}{3}[/latex] with remainder [latex]23[/latex]
We write down the whole part, [latex]2[/latex], and continue with the remainder. There are [latex]23[/latex] left over, so we look at the next place, the [latex]5[/latex]s:
[latex]23 ÷ 5 = \textcolor{red}{4}[/latex] with remainder [latex]\textcolor{red}{3}[/latex]
This leaves us with [latex]3[/latex], which is less than our base, so this number will be in the “ones” place. We are ready to assemble our base-five number:
[latex]348 = (\textcolor{red}{2} × 5^{3}) + (\textcolor{red}{3} × 5^{2}) + (\textcolor{red}{4} × 5^{1}) + (\textcolor{red}{3} × 1)[/latex]
Hence, our base-five number is [latex]\textcolor{red}{2343}[/latex]. We’ll say that [latex]348_{10}=2343_{5}[/latex].